YES, you need a special low voltage fixture for your Airstream.
The existing interior lights in your Flying Cloud are wired for 12-volt DC automotive (or LED) bulbs. The Home Depot fixture is most likely 120-volts AC. (Does the Home Depot fixture plug in the wall at home? If it does, it's wired for 120 VAC.) So, you'll probably need to replace the bulbs and socket(s).
You may be able to cannibalize the parts from the old fixture, and modify them as necessary to fit in the new one. Alternately, you can shop for a lighting fixture at an RV store; e.g., Camping World.